TG EAPCET 2025 (Online) 2nd May Evening Shift

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

The local maximum value $l$ and local minimum value $m$ of $f(x)=\frac{x^2+2 x+2}{x+1}$ in $R-\{-1\}$ exist at $\alpha, \beta$ respectively, then $\frac{l+m}{\alpha+\beta}=$

If $f:[a, b] \rightarrow[c, d]$ is a continuous and strictly increasing function, then $\frac{d-c}{b-a}$ is

If a particle is moving in a straight line so that after $t$ seconds its distance $S$ (in cms) from a fixed point on the line is given by $S=f(t)=t^3-5 t^2+8 t$, then the acceleration of the particle at $t=5 \mathrm{sec}$ is (in $\mathrm{cm...

$P(5,2)$ is a point on the curve $y=f(x)$ and $\frac{7}{2}$ is the slope of the tangent to the curve at $P$. The area of the triangle (in sq. units) formed by the tangent and the normal to the curve at $P$ with $X$-axis is

The area of the region bounded by the curves $y=x^3, y=x^2$ and the lines $x=0$ and $x=2$ is

The constant term in the expansion of $\left(1+\frac{1}{x}\right)^{20}\left(30 x(1+x)^{29}+(1+x)^{30}\right)$ is

When $|x|>3$, then coefficient of $\frac{1}{x^n}$ in the expansion of $x^{3 / 2}(3+x)^{1 / 2}$ is

The slope of a common tangent to the circles $x^2+y^2=16$ and $(x-9)^2+y^2=16$ is

Suppose $C_1$ and $C_2$ are two circles having no common points, then

The centres of all circles passing through the points of intersection of the circles $x^2+y^2+2 x-2 y+1=0$ and $x^2+y^2-2 x+2 y-2=0$ and having radius $\sqrt{14}$ lie on the curve

The locus of the centre of the circle touching the $X$-axis and passing through the point $(-1,1)$ is

The equation of the circle whose radius is 3 and which touches the circle $x^2+y^2-4 x-6 y-12=0$ internally at $(-1,-1)$ is

$A$ circle $S$ given by $x^2+y^2-14 x+6 y+33=0$ cuts the $X$-axis at $A$ and $B(O B>O A)$. $C$ is mid-point of $A B . L$ is a line through $C$ and having slope ( -1 ). If $L$ is the diameter of a circle $S^{\prime}$ and also the radical axi...

In argand plane, no value of $\sqrt[3]{1-i \sqrt{3}}$ lie in

$\omega$ is a complex cube root of unity and $Z$ is a complex number satisfying $|Z-1| \leq 2$. The possible values of $r$ such that $|Z-1| \leq 2$ and $\left|\omega Z-1-\omega^2\right|=r$ have no common solution are

If $|Z|=2, Z_1=\frac{Z}{2} e^{i \alpha}$ and $\theta$ is the $\operatorname{amp}(Z)$, then $\frac{Z_1^n-Z_1^{-n}}{Z_1^n+Z_1^{-n}}=$

If $n, K \in N$ such that $n \neq 3 K$, then $(\sqrt{3}+i)^{2 n}+(\sqrt{3}-i)^{2 n}=$

Let $m, n, p, q$ be four positive integers. If
$$ \begin{aligned} & \int_0^{2 \pi} \sin ^m x \cos ^n x d x=4 \int_0^{\pi / 2} \sin ^m x \cos ^n x d x \int_0^{2 \pi} \sin ^p x \cos ^n x d x=0 \\ & \int_0^\pi \sin ^p x \cos ^q x d x=0, a=m+n+...

\(\mathop {\lim }\limits_{n \to \infty } \frac{1}{n^2}\left[e^{1 / n}+2 e^{2 / n}+3 e^{3 / n}+\ldots+2 n e^2\right]=\)

The equation which represents the system of parabolas whose axis is parallel to $Y$-axis satisfies the differential equation.

The substitution required to reduce the differential equation $t^2 d x+\left(x^2-t x+t^2\right) d t=0$ to a differential equation which can be solved by variables separable method is

If $\frac{d}{d x}\left\{\left(\frac{x-1}{x-\sqrt{x}}\right) e^{2 x+1}\right\}=\frac{x-1}{x-\sqrt{x}} e^{2 x+1} f(x)$, then $f(4)=$

If any tangent drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ touches one of the circles $x^2+y^2=\alpha^2$, then the range of $\alpha$ is

A line segment joining a point $A$ on $X$-axis to a point $B$ on $Y$-axis is such that $A B=15$. If $P$ is a point on $A B$ such that $\frac{A P}{P B}=\frac{2}{3}$, then the locus of $P$ is

If $f: R-\{0\} \rightarrow R$ is defined by $3 f(x)+4 f\left(\frac{1}{x}\right)=\frac{2-x}{x}$ then $f(3)=$

The inverse of the function $y=\frac{10^x-10^{-x}}{10^x+10^{-x}}+1$ is $x=$

The curve represented by $\frac{x^2}{12-\alpha}+\frac{y^2}{\alpha-10}=1$ is

Let $x$ be the eccentricity of a hyperbola whose transverse axis is twice its conjugate axis. Let $y$ be the eccentricity of another hyperbola for which the distance between the focii is 3 times the distance between its directrices. Then $y...

If $l$ is the maximum value of $-3 x^2+4 x+1$ and $m$ is the minimum value of $3 x^2+4 x+1$, then the equation of the hyperbola having foci at $(l, 0),(7 m, 0)$ and eccentricity as 2 is

$\int \frac{x^3}{x^4+3 x^2+2} d x=$

If $I_n=\int \frac{1}{\left(x^2+1\right)^n} d x$, then $2 n I_{n+1}-(2 n-1) I_n=$

\(\int\left(\frac{1}{x^2}+\frac{\sin ^3 x+\cos ^3 x}{\sin ^2 x \cos ^2 x}\right) d x=\)

If $\frac{x^2-3}{(x+2)\left(x^2+1\right)}=\frac{A}{x+2}+\frac{B x+C}{\left(x^2+1\right)}$, then $3 A+2 B-C=$

If $\int \frac{d x}{\left(x^2+9\right) \sqrt{x^2+16}}=\frac{1}{3 \sqrt{7}} \tan ^{-1}\left(K \frac{x}{\sqrt{16+x^2}}\right)+c$, then $K=$

Consider the following statements
Assertion (A) : When $x, y, z$ are positive numbers, then
$$ \begin{aligned} & \tan ^{-1}\left(\sqrt{\frac{x(x+y+z)}{y z}}\right)+\tan ^{-1}\left(\sqrt{\frac{y(x+y+z)}{x z}}\right) +\tan ^{-1}\left(\sqrt{\f...

If $e^{\left(\sinh ^{-1} 2+\cosh ^{-1} \sqrt{6}\right)}=(a+(b+\sqrt{c}) \sqrt{a}+b \sqrt{c})$, then $a+b+c=$

If $y=\left(\sin ^{-1} x\right)^2$, then $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}=$

Consider the following statements
Assertion (A) For $x \in R-\{1\}$;
\(\frac{d}{d x}\left(\tan ^{-1}\left(\frac{1+x}{1-x}\right)\right)=\frac{d}{d x}\left(\tan ^{-1} x\right)\)
Reason (R) For $x<1, \tan ^{-1}\left(\frac{1+x}{1-x}\right)=\...

If $f(x)=\frac{x\left(a^x-1\right)}{1-\cos x}$ and $g(x)=\frac{x\left(1-a^x\right)}{a^x\left(\sqrt{1-x^2}-\sqrt{1+x^2}\right)}$, then $\lim _{x \rightarrow 0}(f(x)-g(x))=$

If the function $g(x)=\left\{\begin{array}{cl}K \sqrt{x+1} & , 0 \leq x \leq 3 \\ m x+2 & , 3 < x \leq 5\end{array}\right.$ is differentiable, then $K+m=$

If $f(x)=\left\{\begin{array}{cc}\frac{a \sin x-b x+c x^2+x^3}{2 \log (1+x)-2 x^3+x^4} & , x \neq 0 \\ 0 & , x=0\end{array}\right.$
is continuous at $x=0$, then

$A, C$ are $3 \times 3$ matrices $B, D$ are $3 \times 1$ matrices. If $A X=B$ has unique solution and $C X=D$ has infinite number of solutions, then

$A$ and $B$ are two non-square matrices. If $P=A+B, Q=A^T B, R=A B^T$, then the matrices whose order is equal to the order of $A$ are

If $\left|\begin{array}{ccc}1 & 2 & 3-\lambda \\ 0 & -1-\lambda & 2 \\ 1-\lambda & 1 & 3\end{array}\right|=A \lambda^3+B \lambda^2+C \lambda+D$, then $D+A=$

If $A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]$ and $2 A-B=\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right]$, then $\operatorname{tr}(A)-\operatorname{tr}(B)=$

For the parabola $y=x^2-3 x+2$, match the items in List I to that of the items in List II. $S$ is a focus, $Z$ is intersection of axis and directrix, $P$ is one end of latus rectum, $Q$ is the point on the parabola at which tangent is paral...

The locus of a point which divides the line segment joining the focus and any point on the parabola $y^2=12 x$ in the ratio $m: n(m+n \neq 0)$ is a parabola.
Then, the length of the latus rectum of that parabola is

The number of integers lying between 1000 and 10000 such that the sum of all the digits in each of those numbers becomes 30 is

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2, z \geq 5$ is